ABSENCE OF CONTINUOUS SPECTRAL TYPES FOR CERTAIN
NONSTATIONARY RANDOM MODELS
IN MEMORY OF ROBERT M. KAUFFMAN
ANNE BOUTET DE MONVEL∗ , PETER STOLLMANN† , AND GUNTER STOLZ‡
Abstract. We consider continuum random Schrödinger operators of the type Hω =
−∆ + V0 + Vω with a deterministic background potential V0 . We establish criteria for
the absence of continuous and absolutely continuous spectrum, respectively, outside
the spectrum of −∆ + V0 . The models we treat include random surface potentials
as well as sparse or slowly decaying random potentials. In particular, we establish
absence of absolutely continuous surface spectrum for random potentials supported
near a one-dimensional surface (“random tube”) in arbitrary dimension.
1. Introduction
In this article we are concerned with spectral properties of certain nonstationary
random models. This type of models has attracted considerable interest as it allows
to study a transition from pure point to continuous spectrum. Here, we are mainly
concerned with the former phenomenon. We obtain our results by essentially “deterministic” techniques from [27, 22, 28] which gives us considerable flexibility. In
particular, we are able to avoid some of the typical technical restrictions that come
with the usual multiscale analysis or fractional moments proofs of localization. E.g.,
we can allow for perturbations of changing sign and single site distributions without
any continuity. On the other hand, we need decaying randomness in the sense that
near infinity the random perturbation is not too effective. That excludes identically
distributed random parameters in most cases. An important exception is our result on
1-D “surfaces” (rather tubes) in arbitrary dimensions.
The paper is organised in the following way: In Section 2 we present the techniques
we use, recalling the relevant notions and results from [27, 22, 28]; in fact we will need
results that are a little stronger than what is explicitly stated in the above cited articles. The common flavour of these methods is that they provide comparison criteria
for the absence of continuous and absolutely continuous spectra, respectively. These
criteria are formulated in the following way: We consider Schrödinger operators with
two potentials that differ only on a set that is “small near infinity in a certain geometrical sense”. Then the spectrum of the first operator has no absolutely continuous
component on the resolvent set of the second one. To exclude continuous spectrum one
needs a bit more complicated assertions involving randomization.
In Section 3 we are dealing with sparse random potentials. The framework we
introduce is fairly general and includes as special cases the sparse random models
considered in [7], e.g. random scatterers are distributed quite arbitrarily in space and
the single site perturbations are assumed to be picked with probabilities that tend to
zero near infinity. Then, throughout the resolvent set of the unperturbed operator there
is no absolutely continuous spectrum. Since we can treat quite general unperturbed
1
2
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
operators, this includes cases with gaps in the spectrum of the unperturbed operator,
a case that is completely new. In the proof we combine elementary combinatorial
arguments, Lemma 3.2, with the methods discussed above. In the same fashion, under
a bit more incisive conditions concerning the background and at least one random
scatterer but with the same condition concerning the decay of probabilities near infinity,
we can even deduce absence of continuous spectrum outside the resolvent set of the
unperturbed operator. This is quite different from what one can obtain with the usual
localization proofs. These proofs require some disorder condition, or apply to energies
near the gaps only (with the exception of the one-dimensional case).
In Section 4 we study surface-like structures. This means we consider potentials
that are concentrated near a subset of lower dimension. Our strongest result, Theorem
4.1 concerns what we call quasi-1D surfaces. There is quite some literature on surface
potentials. Most are dealing with the discrete case [4, 5, 8, 9, 11, 10, 13, 14] while in
[3, 7] and the present paper continuum models are treated. Here again, our goal was to
be able to exclude absolutely continuous spectrum on all of the unperturbed resolvent
set.
In the last section we conclude with a discussion of some possible extensions of our
results and a comparison with other works, in particular the results in [10] and [7].
2. Comparison criteria for absence of (absolutely) continuous spectrum
In this section we present our methods of proof, essentially taken from [27, 22, 28].
These methods rely on comparison of the spectral properties of Schrödinger operators
H1 = −∆ + V1 and H2 = −∆ + V2
whose “difference” is “small” in the sense that the set
{V1 6= V2 } := {x ∈ Rd | V1 (x) 6= V2 (x)}
is sufficiently sparse. To this end we introduce the following concept, following [27]:
Definition. A sequence (Sn )n∈N of compact subsets of Rd with Lebesgue measure
|Sn | = 0 (n ∈ N) is called a total decomposition if there exists a family (Ui )i∈I of
disjoint, open, bounded sets such that
[
[
Rd \
Sn =
Ui .
n∈N
i∈I
A typical example would be Sn = ∂B(0, n), where B(x, r) denotes the closed ball of
radius r, centered at x. (Let us stress that the Sn ’s need not be pairwise disjoint.)
The sparseness of {V1 6= V2 } will be expressed by the existence of a total decomposition (Sn )n∈N with sufficient distance of Sn to {V1 6= V2 } compared with the size of Sn .
An appropriate notion of size is given by the generalized surface area of a set, a notion
introduced in [22] in the following way; here S ⊂ Rd is compact:
σ(S) := sup
r≥0
|{x ∈ Rd | r ≤ dist(x, S) ≤ r + 1}|
.
rd + 1
It is easily seen that
σ(S) ≤ C ((diam S)d + 1),
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
3
i.e. σ(S) is at worst a volume, while for sufficiently regular surfaces it is a surface area
measure, for example
σ(∂B(x, r)) ≤ C(rd−1 + 1).
We cite the following result, essentially taken from [27]:
2.1. Theorem. Assume that for each γ > 0 there exists a total decomposition (Sn )n∈N =
(γ)
(Sn )n∈N such that
δn = δn(γ) := dist({V1 6= V2 }, Sn ) → ∞ as n → ∞
(2.1)
and
X
σ(Sn )e−γδn < ∞.
(2.2)
n
Then
σac (H1 ) ∩ %(H2 ) = ∅.
Here, and in what follows, all potentials V are assumed to be locally uniformly in
Lp , where p ≥ 2 if d ≤ 3 and p > d/2 if d > 3, i.e.
Z
p
kV kp,unif := sup
|V (y)|p dy < ∞.
(2.3)
x
B(x,1)
Theorem 2.1 is essentially Theorem 4.2 from [27]. We will need the slightly stronger
version provided above in which the decomposition Sn may vary with γ. The proof
provided in [27] goes through under this weaker assumption. This is roughly seen as
follows: It suffices to show that
σac (H1 ) ∩ J = ∅
(2.4)
for all compact subsets J of %(H2 ). For fixed J the argument in [27] provides an
γ > 0 (roughly the exponential decay rate in a Combes-Thomas type bound on the
resolvent of H2 for energies in J) such that the validity of (2.1) and (2.2) for a suitable
decomposition will imply (2.4).
Also, in [27] all potentials are assumed to have locally integrable positive parts and
negative parts in the Kato class. Our Lp -type assumptions are a special case.
The second result we use is taken from [28] and excludes continuous spectrum. It is
clear that a statement of the form of Theorem 2.1 above has to be false, since dense pure
point spectrum is extremely instable and can be destroyed by “tiny” perturbations [26].
The geometry is somewhat similar to what we had above but more restrictive.
Namely,
S
consider an increasing sequence (An )n∈N of bounded open sets with n An = Rd . Then
Sn := ∂An is a total decomposition. For the arguments in [28] it is not necessary that
|∂Sn | = 0, but this will be the case in all our applications.
We assume that
δn0 := min{dist(Sn , {V1 6= V2 }), 21 dist(Sn , Sn−1 ∪ Sn+1 )} > 0.
d+1
2.2. Theorem. Assume that V1 ∈ Lloc2 (Rd ), W ∈ L∞ with compact support, of fixed
sign and such that |W | ≥ cχB(0,s) for suitable c > 0 and s > 0. Moreover, assume that
for every γ > 0 there exist An = An (γ) as above such that δn0 = δn0 (γ) → ∞ and
X
0
|An+1 \ An−1 | e−γδn < ∞.
(2.5)
n
4
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
Then for the family Hλ := H1 + λW , λ ∈ R there exists a measurable subset M0 ⊂ R
such that |R \ M0 | = 0 and
σc (Hλ ) ∩ %(H2 ) = ∅ for all λ ∈ M0 .
See [28] for the proof which extends to the case of W as specified above.
Again, as with Theorem 2.1 above, the possible γ-dependence of the sets An is not
explicitly stated in [28], but allowed for by the proof provided there.
The requirement that the summability conditions (2.2), (2.5) have to hold for all
γ > 0 (and suitable decompositions) come from the fact that we want to exclude
(absolutely) continuous spectrum up to the edges of σ(H2 ). It is possible to quantify
and refine the results in a way which says that validity of (2.2), (2.5) for a fixed γ implies
absence of (absolutely) continuous spectrum in regions above a certain (γ-dependent)
distance from σ(H2 ).
3. Sparse random models
In this section we will show how to use the methods from the preceding section
to prove absence of continuous or absolutely continuous spectrum for sparse random
potentials. As mentioned in the introduction, these models have been set up to study
situations in which a transition from singular to absolutely continuous spectrum occurs.
This has attracted some interest in the last decades as can be seen in the articles
[15, 16, 19, 20, 21, 23, 24] dealing with discrete Schrödinger operators and [7] for the
continuum case.
We will be concerned mainly with absence of a continuous spectral component away
from the spectrum of the unperturbed operator. For this reason we state our results
in a generality that does include cases in which no absolutely continuous spectrum
survives. As model examples, let us mention two families of models that have been
treated in [7].
Throughout, the single site potentials will be assumed to be compactly supported
and in Lp , p as above.
In fact, it will be sufficiently interesting to think of compactly supported, bounded
f as done in [7]. However, we shall not restrict ourselves to a fixed sign of f .
Specific models of sparse random potentials, as considered in [7], are
Model 1.
Vω (x) =
X
ξi (ω)f (x − i),
ω∈Ω
i∈Zd
where the ξi are independent Bernoulli variables. Set pi := P(ξi = 1). If pi → 0 as
|i| → ∞ the random potential will no longer be stationary. In fact, it will be sparse in
the sense that almost surely large islands near ∞ will occur where Vω vanishes.
For the second model the ξi and pi will have the same meaning and, additionally,
the qi are i.i.d. nonnegative random variables.
Model 2.
Vω (x) =
X
qi (ω)ξi (ω)f (x − i).
i∈Zd
Again, Vω is sparse in the above sense. Of course, for pi ≡ 1 we would get the usual
Anderson model. Hundertmark and Kirsch study in [7] the metal insulator transition
for H(ω) = −∆ + Vω in L2 (Rd ) for the case that pi → 0 as |i| → ∞ but not too fast in
order to make sure that σess (H(ω)) ∩ (−∞, 0) 6= ∅.
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
5
In the following we consider:
(A1 ) V0 : Rd → R which is locally uniformly Lp with p ≥ 2 if d ≤ 3 and p > d/2 if
d > 3.
(A2 ) Σ ⊂ Rd a set of sites that is uniformly discrete in the sense that
inf{|j − i| | j, i ∈ Σ, j 6= i} =: rΣ > 0.
(A3 ) For each i ∈ Σ a single site potential fi ∈ Lp such that, for finite constants ρ
and M ,
supp fi ⊂ B(0, %) and kfi kp ≤ M.
(A4 )
X
ωi fi (x − i)
i∈Σ
N
(RΣ , i∈Σ µi ), i.e.
Vω (x) =
where ω = (ωi )i∈Σ ∈ (Ω, P) =
the ωi are independent random variables with distribution µi , and supp µi ⊂ [0, 1] for all i ∈ Σ.
For our results on absence of continuous spectrum, in order to apply Theorem 2.2,
we will also require
(d+1)/2
(A5 ) Let V0 , fi ∈ Lloc
(Rd ) for all i ∈ Σ. There exists one k ∈ Σ with fk of
definite sign, bounded, and such that |fk | ≥ cχB(0,s) for some c > 0 and s > 0.
For further reference denote
pi (ε) := µi ([ε, 1]) = P{ωi ≥ ε}.
(3.1)
mk := (µk )ac ([0, 1]),
(3.2)
Also, denote by
the total mass of the absolutely continuous component (µk )ac of µk . We will only use
this for the fixed k ∈ Σ given in (A5 ).
We consider the self-adjoint random Schrödinger operator
H(ω) = H0 + Vω in L2 (Rd )
(3.3)
where H0 = −∆ + V0 . Our assumptions guarantee that the local Lp -bounds (2.3) for
V0 + Vω are uniform not only in x, but also in ω.
Of course, our model contains models I and II above as special cases and pi (ε) ≤ pi
for any ε > 0 in these cases. We have the following result:
3.1. Theorem. Let H(ω) be as above, satisfying (A1 ) to (A4 ), and assume that for all
ε > 0,
pi (ε) = o(|i|−(d−1) ) as |i| → ∞.
(3.4)
Then
(a) σac (H(ω) ∩ %(H0 ) = ∅ almost surely.
(b) Assume, moreover, that (A5 ) holds. Then, with k as in (A5 ),
P{σc (H(ω)) ∩ %(H0 ) = ∅} ≥ mk .
(3.5)
In particular, σc (H(ω)) ∩ %(H0 ) = ∅ holds almost surely if µk is purely absolutely
continuous, without any assumption on the distribution at the other sites.
In order to apply the results from Section 2 we need to find sufficiently many and
sufficiently large regions in which the random potential Vω is small and thus Hω close
to H0 . We start by showing that these regions appear with probability one.
Call a set U ε-free for ω if ωi ≤ ε for all i ∈ Σ ∩ U .
6
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
Denote by
Ar,R = B(0, R) \ B(0, r)
(3.6)
the annulus with inner radius r and outer radius R.
3.2. Lemma. Fix ε > 0 and a > 1. For n ∈ N let
an := P Ar,r+n is not ε-free for all r ∈ [an , an+1 − n] .
(3.7)
Then Σn an < ∞.
Proof. Choose η > 0 such that a(1 − η) > 1. Using uniform discreteness of Σ we get
that for all n ∈ N and r ≥ 1,
#(Ar,r+n ∩ Σ) ≤ Cnrd−1 ,
(3.8)
where C depends on d and rΣ . Here # A is the cardinality of a set A. With C from
(3.8) choose δ ∈ (0, η/(Cad−1 )).
By (3.4), pi (ε) ≤ δ|i|−(d−1) for i sufficiently large. Thus, for sufficiently large n and
each r ∈ [an , an+1 − n],
Y
P(Ar,r+n is ε-free ) =
(1 − pi (ε))
i∈Ar,r+n ∩Σ
≥ (1 − δ|i|−(d−1) )#(Ar,r+n ∩Σ)
≥ (1 − δa−n(d−1) )Cna
≥ (1 − Cδad−1 )n
(n+1)(d−1)
≥
(1 − η)n .
(3.9)
Aan ,an+1 contains at least n1 (an+1 − an ) − 1 disjoint annuli Aj := Arj ,rj +n of width
n. Thus, using independence and (3.9),
an ≤ P(no Aj is ε-free)
Y
=
P(Aj is not ε-free )
j
−1 (an+1 −an )−1
≤ (1 − (1 − η)n )n
≤ e−(1−η)
n (n−1 an (a−1)−1)
.
As (1 − η)a > 1, the an are summable.
(3.10)
By the Borel-Cantelli lemma we conclude P(Ωε,a ) = 1, where
Ωε,a := ω ∈ Σ : For each sufficiently large n the annulus Aan ,an+1
contains a sub-annulus Arn ,rn +n which is ε-free for ω .
(3.11)
Therefore
Ωε =
\
Ωε,1+1/`
(3.12)
`∈N
also has full measure.
Based on this we can now complete the
Proof of Theorem 3.1. Fix a compact K ⊂ %(H0 ). Since %(H0 ) can be exhausted by
an increasing sequence of compact subsets, it suffices to prove that
σac (H(ω)) ∩ K = ∅ almost surely.
(3.13)
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
7
It can be shown, using the general theory of uniformly local Lp potentials, e.g. [25],
that there is an ε0 > 0 such that
σ(H0 + V ) ∩ K = ∅,
(3.14)
ε0 .
for each V with kV kp,unif ≤
Thus, by the properties of Σ and fi , there is an ε > 0
such that
X
σ(H0 +
δi fi (x − i)) ∩ K = ∅
(3.15)
i∈Σ
if |δi | ≤ ε for all i ∈ Σ.
Fix this ε > 0 and let Ωε be the full measure set found above. For given ω ∈ Ωε let
ω̃i := min{ωi , ε}, i ∈ Σ, and
X
V2 (x) :=
ω̃i fi (x − i).
i∈Σ
By (3.15) we have σ(H0 + V2 ) ∩ K = ∅. Thus, in order to apply Theorem 2.1 and
(γ)
conclude that (3.13), it suffices to find for every γ > 0 a total decomposition (Sn ) of
{Vω 6= V2 } which satisfies (2.1) and (2.2).
For given γ > 0 choose an integer ` > 2(d − 1)/γ. This implies (d − 1) log a < γ/2,
where a := 1 + 1/`. As ω ∈ Ωε,a , for each sufficiently large n the annulus Aan ,an+1
contains an ε-free annulus Arn ,rn +n .
(γ)
Choose Sn := ∂B(0, rn + n2 ). Then
n
δn(γ) := dist({Vω 6= V2 }, Sn(γ) ) ≥ − ρ
2
(γ)
since Arn ,rn +n is ε-free (recall that supp fk ⊂ B(0, ρ)). Thus δn → ∞. Also using
(γ)
that σ(Sn ) ≤ Can(d−1) , we conclude
X
X
(γ)
σ(Sn(γ) )e−γδn ≤ Ceγρ
en((d−1) log a−γ/2) < ∞.
n
n
This proves part (a) of Theorem 3.1.
In order to apply Theorem 2.2 to prove part (b) we slightly modify the above construction, essentially replacing Σ by Σ \ {k}.
Let Ω0 := RΣ\{k} with measure P0 = ⊗i∈Σ\{k} µi . As the property defining Ωε,a in
(3.11) does not depend on the value of ωk , we get that also P0 (Ω0ε,a ) = P0 (Ω0ε ) = 1,
where Ω0ε,a and Ω0ε are defined as in (3.11) and (3.12), but as subsets of Ω0 .
For compact K ⊂ %(H0 ) choose ε > 0 as in the proof
of part (a). For ω 0 ∈ Ω0ε let
P
0
0
ω̃i := min{ωi , ε} (i ∈ Σ \ {k}). Also let Vω0 (x) = i∈Σ\{k} ωi0 fi (x − i) and V20 (x) =
P
0
0
i∈Σ\{k} ω̃i fi (x − i). As before, σ(H0 + V2 ) ∩ K = ∅.
For γ > 0 choose ` > 2d/γ, a = 1 + 1/`. With rn from (3.11), let An = B(0, rn + n2 )
and Sn = ∂An . This yields
|An+1 \ An−1 | ≤ cd a(n+2)d
and
n
1
= min dist(Sn , {V 6= V2 }), dist(Sn , Sn−1 ∪ Sn+1 ) ≥ − ρ.
2
2
P
0
−γδ
The choice of a guarantees that n |An+1 \ An−1 |e n < ∞. By Theorem 2.2 this
proves the existence of a measurable subset M0,ω0 ⊂ R with |R \ M0,ω0 | = 0 and such
that
σc (H(λ, ω 0 ) ∩ K) ⊂ σc (H(λ, ω 0 ) ∩ ρ(H0 + V20 )) = ∅
δn0
ω0
8
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
for all λ ∈ M0,ω0 , where H(λ, ω 0 ) = H0 + λfk (x − k) + Vω0 (x).
As µk (M0,ω0 ) ≥ (µk )ac (M0,ω0 ) = (µk )ac (R) = mk it follows by Fubini that P{ω ∈ Ω :
σc (H(ω)) ∩ K = ∅} ≥ mk . Since this bound is independent of K and we can exhaust
ρ(H0 ) by an increasing sequence Kn we arrive at the assertion. This completes the
proof of Theorem 3.1.
Remark. While the “volume” term |An+1 \ An−1 | in (2.5) has to be considered larger
than the “surface” term σ(Sn ) in (2.2), this did not make a significant difference in
the above proof. The same total decomposition Sn can be used to prove absence of
absolutely continuous spectrum and absence of continuous spectrum. The difference
will become more significant for the quasi-1D surfaces considered in the next section.
The reader will have noticed that the choice of a polynomial bound in the assertion is
somewhat arbitrary. In fact, what we use is a Metatheorem of the form that the almost
sure appearance of ε-free annular regions allows one to exclude absolutely continuous
spectrum outside the unperturbed %(H0 ). If some more regularity holds for one of the
coupling constants, then one can even conclude that the spectrum is pure point outside
%(H0 ) with positive probability.
Of course, the appearance of suitable geometries might also be forced by the distribution of Σ in space: think of Σ that is sparse near infinity.
Remarks. (1) Note that in (a) of Theorem 3.1 we can deal with indefinite single site
potentials and quite arbitrary single site distributions.
(2) Of course, the assumption supp µj ⊂ [0, 1] and 0 ∈ supp µj is just to normalize
things. For our methods to apply the random potentials have to obey some uniform
bounds.
(3) In the special case of model II considered by Hundertmark and Kirsch our result
is stronger for d = 1, and weaker for d ≥ 2. However, since they announce a proof by
multiscale analysis, it is not obvious how they want to exclude continuous spectrum
for all negative energies. Typically, multiscale techniques only work near band edges.
Most probably, they implicitly have a disorder or large coupling assumption in mind.
Let us stress that our methods of proof work for all energies outside %(H0 ).
(4) In the special case of model I considered in [7] the negative part of the spectrum
is purely discrete and this cannot support continuous components. Out more general
results, however, apply in situations where the spectrum outside %(H0 ) fills an interval
in the negative reals.
Using the “Almost surely free lunch Theorem” from [7] we get the following result
for V0 = 0:
3.3. Theorem. Let µk , fk , V0 be as above, V0 = 0 and assume that, additionally, the
kfk k∞ are uniformly bounded and that the second moments of the ηk obey
Z 1
E(ηk2 ) =
x2 dµk ≤ C|k|−β
0
for some β > 2. Then
σac (H(ω)) ⊃ [0, ∞) P -a.s.
Proof. The assumptions clearly make sure that
1
W (x) := E(Vω (x)2 ) 2 ≤ C(1 + |x|)−(1+ε)
so that we can apply Theorem 2.4 from [7] to see that Cook’s criterion is applicable for
P -a.e. ω ∈ Ω.
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
9
For general V0 the corresponding ist probably false. It should be true for certain
periodic potentials, see [2, 6, 29].
4. Quasi-1D surfaces
In Section 3 sparseness of the potential Vω in (A4 ) resulted from an assumption on
decaying randomness, e.g. (3.4). In the present section we will modify our methods
and results for the case where sparseness of Vω arises directly through sparseness of
the deterministic set Σ. By this we mean situations where Σ does not have positive
d-dimensional density in Rd , i.e. #(Σ ∩ B(0, R)) = o(Rd ) as R → ∞. A special case
would be an m-dimensional sublattice, e.g. Σ = Zm × {0} ⊂ Rm × Rd−m , 0 < m < d,
in which case Vω would model a random surface potential. Our most interesting result
holds for m = 1, where our methods cover the following more general situation:
Definition. A uniformly discrete subset Σ of Rd is called quasi-one-dimensional (quasi1D) if there exists C < ∞ such that
#(Σ ∩ AR,R+1 ) ≤ C
(4.1)
for all R ≥ 0.
4.1. Theorem. Let H(ω) = H0 + Vω satisfy (A1 ) to (A4 ). In addition, assume that Σ
is quasi-1D and that
sup pi (ε) < 1
(4.2)
i∈Σ
for every ε > 0. Then σac (H(ω)) ∩ %(H0 ) = ∅ almost surely.
If Σ is quasi-1D, then by Theorem 4.1, no spatial decay in the randomness of the
ηi is required to conclude absence of absolutely continuous spectrum in gaps of σ(H0 ).
For example, (4.2) is satisfied for independent, identically distributed random variables
ηi such that 0 ∈ supp µ for their common distribution µ. In particular, as every
uniformly discrete Σ ⊂ R is quasi-1D, this strengthens Theorem 3.1(a) in the case
d = 1, which would require pi (ε) = o(1) as k → ∞. Of course, in the case d = 1 our
result is hardly new as (essentially) much stronger results are known for one-dimensional
random potentials.
More interesting is the case d > 1, where special cases of quasi-1D sets include
discrete tubes of the form Σ = Z × S, with S a bounded subset of Zd−1 . Theorem 4.1
shows the absence of absolute continuity in the “surface spectrum” generated by the
random (1D) surface potential V (ω). Also, within certain limitations, we can allow for
curvature in the tubes Σ, thus covering rather general “random sausages”.
One can find quite a number of results concerning “corrugated” or “random” surfaces.
Most are concerned with discrete models; see [4, 5, 8, 9, 11, 10, 13, 14], and [3, 7] for
continuoum variants. Most reminiscent of what we have here, in [10] the authors
present a result stating that the spectrum induced by a oonedimensional surface in
discrete twodimensional space is almost surely pure point outside the spectrum [−4, 4]
of the unperturbed operator. The proof, however, is pretty much involved and not all
cases are worked out in detail.
Proof. We start with a modification of Lemma 3.2.
4.2. Lemma. Fix ε > 0. Let δ = supi pi (ε) < 1, C as in (4.1) and a >
an , as defined in (3.7), is summable.
1
.
(1−δ)C
Then
10
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
Proof of Lemma. This follows with the same argument as in the proof of Lemma 3.2,
using that now P(Ar,r+n is ε-free ) ≥ (1 − δ)Cn . Thus the set Ωε,a , defined as in (3.11),
has full P-measure.
Fix K ⊂ %(H0 ) compact and argue as in the proof
P of Theorem 3.1 to find ε > 0
such that σ(H0 + V2 ) ∩ K = ∅, where V2 (x) =
i∈Σ ω̃i fi (x − i), ω̃i = min{ωi , ε}.
Choose a > 1 as in Lemma 4.2 and ω ∈ Ωε,a , i.e. Aan ,an+1 contains ε-free Arn ,rn +n for
all sufficiently large n.
As before, the spheres Sn = ∂B(0, rn + n2 ) give a total decomposition with dist({Vω 6=
V2 }, Sn ) ≥ n2 − ρ. But, as Lemma 4.2 prevents us from choosing a arbitrarily close to
1, this will not yield convergence of (2.2) for all γ > 0. We will therefore refine our
construction by splitting the Sn in two parts. One part is a union of spherical caps for
which, due to points of Σ close to Arn ,rn +n , the distance n2 − ρ from {Vω 6= V2 } can’t be
improved. The second part (the remaining “swiss cheese”) has much bigger distance
to {Vω 6= V2 } and, due to the sparseness of Σ, contains most of Sn . The details of this
construction are as follows:
Fix α > 1. Let
Pn := (Arn −nα ,rn ∪ Arn +n,rn +n+nα ) ∩ Σ
(4.3)
be the points of Σ in the nα -neighborhood of Arn ,rn +n (but outside Arn ,rn +n ). For each
j ∈ Pn define the spherical cap
Sn,j := Sn ∩ B((rn +
n j α
) , n ).
2 |j|
(4.4)
Also let
Sn0 := Sn \
[
Sn,j .
j
Since
Sn0
∪
S
j
Sn,j = Sn , we have that
{Sn,j : n ∈ N, j ∈ Pn } ∪ {Sn0 : n ∈ N}
(4.5)
is a total decomposition of Rd . As above, since Arn ,rn +n is ε-free,
n
δn,j := dist({Vω 6= V2 }, Sn,j ) ≥ − ρ.
(4.6)
2
If x ∈ Sn0 and j ∈ Σ ∩ (Arn ,rn +n )c , then, by elementary geometric considerations,
dist(x, j) ≥ nα for sufficiently large n. Using this and again that Arn ,rn +n is ε-free, we
find
δn0 := dist({Vω 6= V2 }, Sn0 ) ≥ nα − ρ.
(4.7)
From the simple volume bound on the generalized surface area one gets
σ(Sn,j ) ≤ Cndα ,
(4.8)
σ(Sn0 ) ≤ Cadn .
(4.9)
Checking (2.2) for the partition (4.5) amounts to proving that
X
0
σ(Sn0 )e−γδn < ∞
(4.10)
n
and of
XX
n j∈Pn
σ(Sn,j )e−γδn,j < ∞
(4.11)
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
11
for each γ > 0. (4.10) follows from (4.7) and (4.9) since α > 1. (4.11) follows from
(4.6) and (4.8), noting that #Pn ≤ 2nα + 2 since Σ is quasi-1D. From Theorem 2.1 we
conclude σac (H(ω)) ∩ K ⊂ σac (H(ω)) ∩ %(H0 + V2 ) = ∅.
Remark. It is possible to prove Theorem 4.1 under a slightly weaker assumption on
the set Σ, namely that there exists C < ∞ such that
#(Σ ∩ B(0, R)) ≤ CR
(4.12)
for all R ≥ 1. (4.12) is weaker than (4.1) in that it allows the number of points in
Σ ∩ AR,R+1 to be unbounded with respect to R. (4.12) is also somewhat more natural
as it doesn’t depend on the norm used to define B(0, R) nor on the choice of the center
of the ball.
A simple counting argument shows that, under the assumption (4.12), for each annulus of the form Aan ,an+1 most sub-annuli AR,R+n satisfy a bound #(Σ ∩ AR,R+n ) ≤ Cn.
Here “most” means a non-vanishing fraction. One finds sufficiently many disjoint such
annuli to construct ε-free regions as before. Moreover, by an additional counting argument, one argues that most of these annuli do not have more than C 0 nα points of
Σ in their nα -neighborhoods. Based on this one can construct a partition {Sn0 , Sn,j }
as above and carry through the proof. We skip the somewhat tedious details of this
generalization.
We are not able to prove a result like Theorem 3.1(b), i.e. absence of continuous
spectrum in %(H0 ) with positive probability, under the assumptions of Theorem 4.1
(plus (A5 )). For the partition Sn = ∂An , An = B(0, rn + n2 ) the volumes |An+1 \ An−1 |
grow too fast to get validity of (2.5) for all γ > 0. A trick like the introduction of
{Sn0 , Sn,j } as above is not applicable here since in Theorem 2.2 the Sn need to arise as
boundaries of a growing sequence An .
However, if one replaces (4.2) by pi (ε) = o(1) as |i| → ∞ for all ε > 0, then
Lemma 4.2 will hold for any a > 1, which allows for an application of Theorem 2.2
with a γ-dependent choice of the Sn , as in the proof of Theorem 3.1(b). Sparseness of
the random potential is achieved here through a combination of sparseness of Σ and
decaying randomness pi (ε) = o(1), as opposed to Theorem 3.1, where sparseness follows
exclusively from stronger decay pi (ε) = o(|i|−(d−1) ).
In fact, the correlation between the degree of sparseness of Σ and the rate of decay
of pi (ε) can be made more specific. For this, call a uniformly discrete set Σ ⊂ Rd
quasi-m-dimensional (1 ≤ m ≤ d, not necessarily integer) if for some C < ∞ and all
R ≥ 0,
#(Σ ∩ AR,R+1 ) ≤ CRm−1 .
(4.13)
Then the following result is found with the same methods as above:
4.3. Theorem. Let H(ω) satisfy (A1 ) to (A4 ), Σ be quasi-m-dimensional and, for all
ε > 0,
pi (ε) = o(|i|−(m−1) ) as |i| → ∞,
(4.14)
then σac (H(ω)) ∩ %(H0 ) = ∅ almost surely.
If, moreover, (A5 ) holds, then P{Σc (H(ω)) ∩ %(H0 )} ≥ mk .
5. Concluding remarks
Among the known results for discrete surface models, the one most closely related to
Theorem 4.1 above is the result of Jakšić and Molchanov [10]. They consider the discrete
12
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
Laplacian on Z × Z+ with random boundary condition ψ(n, −1) = Vω (n)ψ(n, 0), where
the Vω (n) are i.i.d. random variables. They show that the spectrum outside [−4, 4],
i.e. outside the spectrum of the two-dimensional discrete Laplacian, is almost surely
pure point. This is stronger than our continuum analogue in the sense that we can
only prove absence of absolute continuity outside the spectrum of the deterministic
background operator H0 .
The proof in [10] requires a technical tour de force. The two-dimensional problem
can be reduced to a one-dimensional problem with long range interactions. Anderson
localization for the latter has been proven in [12] with methods based on an approach
developed in [17] (which is also behind Theorem 2.2 above). The one-dimensional
problem depends nonlinearly on the spectral parameter, a difficulty which is resolved
by adapting some ideas from the Aizenman-Molchanov fractional moment method [1].
Our methods are comparatively soft. In particular, they work directly in the multidimensional PDE setting and do not require a reduction to d = 1. One-dimensionality
of the random surface only enters through its probabilistic consequences (Lemma 4.2)
for the frequency of the appearance of ε-free regions, which constitute the “potential
barriers” required in Theorem 2.1.
This makes our methods very flexible. In addition to the extension to continuum
models, they allow for rather general quasi-1D surfaces (e.g. curved tubes, unions of
tubes), work in arbitrary dimension d and allow for the presence of an additional
deterministic background potential V0 . It is possible to adapt our methods to lattice
operators and prove absence of absolutely continuous spectrum outside the spectrum of
the discrete Laplacian for much more general geometries than the half-plane considered
in [10].
Also, our methods can easily be adjusted to work for operators of the type (3.3) on
L2 (Ω), Ω 6= Rd . For example, for H(ω) = −∆ + Vω in L2 ((0, a) × Rd−1 ) with Dirichlet
boundary conditions and Vω given through (A2 ) to (A4 ) with i.i.d. coupling constants
ωi , we would get that σac (H(ω)) ∩ (−∞, 0) = ∅ almost surely. Of course, for this
physically one-dimensional operator (with no bulk space), one would expect the much
stronger result that σc (H(ω)) = ∅. But the corresponding result for discrete strips,
e.g. [18], does not seem to extend easily to the continuum.
Finally, we mention that Hundertmark and Kirsch [7] announce some results on
pure point spectrum for continuum models similar to the ones studied here. They will
use suitable adaptations of multiscale analysis to show that the negative spectrum of
−∆ + Vω is almost surely pure point. Here Vω is either of the type of Model 2 above or
a random potential at the surface of a half space Schrödinger operator. In situations
where the multiscale analysis can be carried out, their results should be stronger than
ours.
Acknowledgement:
Our collaboration has been supported by the University Paris 7 Denis Diderot where
part of this work was done, by the DFG in the priority program “Interacting stochastic
systems of high complexity” and through the SFB 393, as well as through US-NSF
grant no. DMS-0245210.
References
[1] M. Aizenman and S. Molchanov: Localization at large disorder and at extreme energies: An
elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)
ABSENCE OF CONTINUOUS SPECTRUM FOR CERTAIN RANDOM MODELS
13
[2] M. Sh. Birman and D. R. Yafaev: The scattering matrix for a perturbation of a periodic Schrdinger
operator by decreasing potential. (Russian) Algebra i Analiz 6, no. 3, 17–39 (1994); translation in
St. Petersburg Math. J. 6, no. 3, 453–474 (1995)
[3] A. Boutet de Monvel and P. Stollmann: Dynamical localization for continuum random surface
models, Arch. Math., 80, 87–97 (2003)
[4] A. Boutet de Monvel and A. Surkova: Localisation des états de surface pour une classe d’opérateurs
de Schrödinger discrets à potentiels de surface quasi-périodiques, Helv. Phys. Acta 71(5), 459–490
(1998).
[5] A. Chahrour and J. Sahbani: On the spectral and scattering theory of the Schrödinger operator
with surface potential. Rev. Math. Phys. 12, no. 4, 561–573 (2000)
[6] C. Gérard and F. Nier: Scattering theory for the perturbations of periodic Schrdinger operators.
J. Math. Kyoto Univ. 38, no. 4, 595–634 (1998)
[7] D. Hundertmark and W. Kirsch: Spectral theory of sparse potentials, in “Stochastic processes,
physics and geometry: new interplays, I (Leipzig, 1999),” Amer. Math. Soc., Providence, RI, 2000,
pp. 213–238
[8] V. Jakšić and Y. Last: Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12(11), 1465–1503
(2000)
[9] V. Jakšić and Y. Last: Spectral structure of Anderson type hamiltonians, Inv. Math. 141 no. 3,
561–577 (2000)
[10] V. Jakšić and S. Molchanov: On the surface spectrum in dimension two, Helv. Phys. Acta 71(6),
629–657 (1998)
[11] V. Jakšić and S. Molchanov: On the spectrum of the surface Maryland model, Lett. Math. Phys.
45(3), 189–193 (1998)
[12] V. Jakšić and S. Molchanov: Localization for one dimensional long range random hamiltonians,
Rev. Math. Phys. 11, 103–135 (1999)
[13] V. Jakšić and S. Molchanov: Localization of surface spectra, Comm. Math. Phys. 208(1), 153–172
(1999)
[14] V. Jakšić, S. Molchanov and L. Pastur: On the propagation properties of surface waves, in “Wave
propagation in complex media (Minneapolis, MN, 1994),” IMA Math. Appl., Vol. 96, Springer,
New York, 1998, pp. 143–154
[15] W. Kirsch: Scattering theory for sparse random potentials. Random Oper. Stochastic Equations
10, no. 4, 329–334 (2002)
[16] W. Kirsch, M. Krishna and J. Obermeit: Anderson model with decaying randomness: Mobility
edge, Math. Z., 235, 421–433 (2000)
[17] W. Kirsch, S. Molchanov and L. Pastur: One-dimensional Schrödinger operators with high potential barriers. Operator Theory, Adv. Appl. 57, 163–170 (1992)
[18] A. Klein, J. Lacroix, A. Speis: Localization for the Anderson model on a strip with singular
potentials. J. Funct. Anal. 94, no. 1, 135–155 (1990)
[19] M. Krishna: Anderson model with decaying randomness: Existence of extended states. Proc.
Indian Acad. Sci. Math. Sci. 100, no. 3, 285–294 (1990)
[20] M. Krishna: Absolutely continuous spectrum for sparse potentials. Proc. Indian Acad. Sci. Math.
Sci. 103, no. 3, 333–339 (1993)
[21] M. Krishna and K. B. Sinha: Spectra of Anderson type models with decaying randomness. Proc.
Indian Acad. Sci. Math. Sci. 111, no. 2, 179–201 (2001)
[22] I. McGillivray, P. Stollmann and G. Stolz: Absence of absolutely continuous spectra for multidimensional Schrödinger operators with high barriers. Bull. London Math. Soc. 27, no. 2, 162–168
(1995)
[23] S. Molchanov: Multiscattering on sparse bumps. Advances in differential equations and mathematical physics (Atlanta, GA, 1997), Contemp. Math., 217, Amer. Math. Soc., Providence, RI,
1998, pp. 157–181
[24] S. Molchanov and B. Vainberg: Multiscattering by sparse scatterers. Mathematical and numerical
aspects of wave propagation (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000,
pp. 518–522
[25] M. Reed and B. Simon: Methods of Modern Mathematical Physics IV: Analysis of Operators,
Academic Press, New York, 1978
[26] B. Simon: Spectral analysis of rank one perturbations and applications. CRM Proc. Lecture Notes,
8, 109–149 (1995)
14
A. BOUTET DE MONVEL, P. STOLLMANN, AND G. STOLZ
[27] P. Stollmann and G. Stolz: Singular spectrum for multidimensional operators with potential barriers. J. Operator Theory 32, 91–109 (1994)
[28] G. Stolz: Localization for Schrödinger operators with effective barriers. J. Funct. Anal. 146, no.
2, 416–429 (1997)
[29] D. Yafaev: Eigenfunctions of the continuous spectrum for the N -particle Schrdinger operator.
Spectral and scattering theory (Sanda, 1992), 259–286, Lecture Notes in Pure and Appl. Math.,
161, Dekker, New York, 1994.
∗
†
‡
IMJ, case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris, France
Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL
35294, USA